Bayesian inference applied to measurements

``...these problems are classified as probability
of the causes, and are the most interesting of all
from their scientific applications''.

``An effect may be produced by the cause a or by the cause b.
The effect has just been observed. We ask the probability that
it is due to the cause a. This is an à posteriori
probability of cause. But I could not calculate it, if
a convention more or less justified did not tell me in advance
what is the à priori probability for the cause a
to come into play. I mean the probability of this event to
some one who had not observed the effect.''
(Henri Poincaré)